Homogeneous HKT and QKT manifolds

TL;DR

This paper introduces a method to construct a broad class of homogeneous KT, HKT, and QKT manifolds using invariant metrics and Dynkin diagram colorings, providing numerous examples and linking structures via twistor spaces.

Contribution

It presents a novel construction approach for homogeneous KT, HKT, and QKT manifolds based on Dynkin diagram colorings, expanding the catalog of known examples.

Findings

Constructed a large class of homogeneous KT, HKT, and QKT manifolds.

Linked Dynkin diagram colorings to specific geometric structures.

Showed twistor spaces of QKT manifolds admit KT structures.

Abstract

We present the construction of a large class of homogeneous KT, HKT and QKT manifolds, $G/K$, using an invariant metric on $G$ and the canonical connection. For this a decomposition of the Lie algebra of $G$ is employed, which is most easily described in terms of colourings of Dynkin diagrams of simple Lie algebras. KT structures on homogeneous spaces are associated with different colourings of Dynkin diagrams. The colourings which give rise to HKT structures are found using extended Dynkin diagrams. We also construct homogeneous QKT manifolds from homogeneous HKT manifolds and show that their twistor spaces admit a KT structure. Many examples of homogeneous KT, HKT and QKT spaces are given.